Problem 80 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

PROPERTY VALUE A community is laid out as a rectangular grid in relation to two main streets that intersect at the city center. Each point in the community has coordinates \((x, y)\) in this grid, for \(-10 \leq x \leq 10,-8 \leq y \leq 8\) with \(x\) and \(y\) measured in miles. Suppose the value of the land located at the point \((x, y)\) is \(V\) thousand dollars, where $$ V(x, y)=(250+17 x) e^{-0.01 x-0.05 y} $$ Estimate the value of the block of land occupying the rectangular region \(1 \leq x \leq 3,0 \leq y \leq 2\).

Step-by-Step Solution

Answer

The value of the block of land is approximately $518.80 thousand.

Step 1: Understand the Function

The value of land at any point \(x, y\) within the community is given by the function \[ V(x, y)=(250 + 17x) e^{-0.01 x - 0.05 y} \].

Step 2: Define the Region

We need to estimate the value of the block of land within the rectangular region \(1 \leq x \leq 3, 0 \leq y \leq 2\).

Step 3: Set Up the Double Integral

To estimate the total value of the block of land, we set up a double integral over the region: \[ \iint_R V(x, y) \, dx \, dy \]. Here, the region \R\ is defined by \

Key Concepts

land value estimation

Land value estimation is a fundamental concept in property valuation and urban planning. It involves assessing the monetary worth of a given piece of land based on various factors like location, usage, and economic conditions. In mathematical terms, we often use functions to model how land value changes across different locations within a defined space. In our given exercise, the land value at any point \((x, y)\) is described by the function \[ V(x, y) = (250 + 17x)e^{-0.01x - 0.05y} \]. This function takes into account both an intrinsic value \(250 + 17x\) and a decay factor \(e^{-0.01x - 0.05y}\) which models how the value reduces as we move away from the origin (city center). For a goal like estimating the value of a specific rectangular block of land, we use the double integral method over the region of interest.

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to specify points in a plane using two numerical values: the x-coordinate and the y-coordinate. These coordinates give us a unique location in the grid where the land is situated. In our exercise, \(-10 \leq x \leq 10\) and \(-8 \leq y \leq 8\) define the overall layout of the community. We need to find the value of land specifically in the region where \(1 \leq x \leq 3\) and \(0 \leq y \leq 2\). This rectangular region determines our limits of integration when performing the double integral and ensures that we are only calculating the value for the specified area.

double integral setup

Setting up a double integral is essential for estimating the total value across a two-dimensional area. The double integral allows us to sum up infinitesimal values over a region in the x-y plane. For our given problem, to calculate the value of the land within the bounds \(1 \leq x \leq 3\) and \(0 \leq y \leq 2\), we express this mathematically as: \[ \text{Total Value} = \int_{y=0}^{2}\int_{x=1}^{3} (250 + 17x)e^{-0.01x - 0.05y} \,dx \,dy \]. The function inside the integral \( (250 + 17x)e^{-0.01x - 0.05y} \) represents the value \(V(x, y)\) at any point \((x, y)\). The limits of integration define the boundaries of our region. Integrating first with respect to \(x\), and then with respect to \(y\), we cover the entire area of interest and sum up the land values.

exponential functions

Exponential functions are critical in modeling decay or growth processes in various scenarios, including land value estimation. The given land value function \(V(x, y) = (250 + 17x)e^{-0.01x - 0.05y}\) includes an exponential term \(e^{-0.01x - 0.05y}\). This term represents a decay factor that decreases the land value as \(x\) and \(y\) increase. The parameters within the exponent \(-0.01x\) and \(-0.05y\) indicate how rapid the value decreases. Consequently, the decay factor significantly impacts the value calculation because the further you go from the city center, the lesser the value. Understanding how exponential functions work helps in comprehending how location coordinates affect land values in our estimation model.